# Download Arithmetic, Geometry, Cryptography and Coding Theory 2009 by David Kohel, Robert Rolland (ed.) PDF

By David Kohel, Robert Rolland (ed.)

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**Example text**

The conditional branching now contains the same operations by adding dummy operations to equalise the side-channel leakage. The standard Montgomery ladder is highly regular as it computes, for each bit regardless of its value, a doubling and an addition. Our multiplication algorithms are based on an adapted Montgomery ladder. Our four proposed algorithms each compute the same sequence of instructions regardless of the value the bit of the scalar takes. The computations are a ﬁxed pattern unrelated to the bit information of k.

41–67 (2000) [3] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user Language, J. Symbolic Computation 24(3-4), pp. 235–265 (1997) 28 8 WOUTER CASTRYCK AND JOHN VOIGHT `s, Curves of genus two over ﬁelds of even characteristic, [4] G. Cardona, E. Nart, J. Pujola Math. Z. 250(2005), no. 1, pp. 177–201 [5] W. Castryck, J. Voight, On nondegeneracy of curves, Algebra & Number Theory 3(3), pp. 255–281 (2009) [6] N. Elkies, The Klein quartic in number theory, pp. 51–102 in S. ), The eightfold way: the beauty of Klein’s quartic curve, MSRI Publication Series 35, Cambridge University Press, 352 pp.

2. Over any ﬁnite ﬁeld k, all curves C/k of genus at most 3 are nondegenerate, except if k = F2 and C is k-birationally equivalent to C2 , or if k = F3 and C is k-birationally equivalent to C3 . Proof. It remains to show that if C is a nonhyperelliptic curve of genus 3 which can be modeled by a nondegenerate Laurent polynomial f , then it can be modeled by a nondegenerate Laurent polynomial whose Newton polytope is contained in 4Σ, the convex hull of the points (0, 0), (0, 4), and (4, 0). 1]. Applying a Z-aﬃne transformation to the exponent vectors, we may assume that in fact the interior lattice points of Δ(f ) are (1, 1), (1, 2), and (2, 1).